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User blog:B1mb0w/J Function Sandpit J 2
'J Function Sandpit \(J_2\)' The J Function is a work in progress. This sandpit defines a function called \(J_2\) which contains ideas that will be used in the final J Function. Click here for the J Function blog. 'Summary' The \(J_2\) function is a cleaner and more successful version of the \(J_0\) function. It uses one real number "r" as an input parameter. The function can then produce effectively every FGH function with ordinals up to \(\epsilon_0\) in order of increasing size, for any number n. This makes for a useful function for referencing some very large numbers which otherwise need complex notation that is difficult to read or understand. 'Principles' Some principles used to create this function are: Starting assumption: the function makes extensive use of the small integers 0 to 7. All other numbers created by the function will be some instance of the FGH function. Note that this starting range includes some trivial instances of FGH. \(1 = f_0 (0)\) \(2 = f_0 (1) = f_1 (1) = f_{\omega}(1) = f_{\epsilon_0}(1)\) \(3 = f_0 (2)\) \(4 = f_1 (2)\) Generating FGH functions: the first FGH function is this trivial example. \(8 = f_2(2) = f_{\omega}(2)\) From here the function generates FGH functions in the following sequence: \(f_2(2), f_2(3), f_2(4), f_2(5), f_2(6), f_2(7)\) then \(f_2^2(3), f_2^2(4), f_2^2(5), f_2^2(6), f_2^2(7)\) note that only values of n = 2 to 7 are used. The sequence continues for higher nested powers "p" of the FGH function. When p = 2 then n = 2 is not used. It is an illegal entry because n > p in all cases. The sequence for p = 3 is shown next. This illustrates how the value of n is restricted by p. \(f_2^3(3), f_2^3(4), f_2^3(5), f_2^3(6), f_2^3(7)\) In this sequence \(f_2^3(3)\) should be illegal but because it can be represented differently as \(f_{\omega}(3)\) it is accepted in the sequence. For the next sequence of p = 4, it is clear that n = 3 is illegal but n = 4 is accepted. \(f_2^4(4), f_2^4(5), f_2^4(6), f_2^4(7)\) The reason n = 4 is accepted because it can be represented differently as \(f_3(4)\) and this illustrates the general rule for creating these ascending sequences. In this example the ordinal "w" of the FGH function is 3 and n > w and n > p at all times. It is not difficult to see the sequence continuing to \(f_2^6 (7)\) after which it proceeds with: \(f_3(4), f_3(5), f_3(6), f_3(7)\) The ordinal w = 3 and the nested power p = 1. The value of n can then range from 4 to 7 following the same rules as before. The nested power can then be increased to p = 2 and following these rules we reach the FGH function \(f_6^7(7)\) which is accepted because it can be represented as \(f_{\omega}(7)\). The sequence at this point has exhausted all the FGH functions using small numbers 0 to 7. We can now use the first generated instance of FGH function, namely \(f_2 (2)\), and replace n = 7 with this function. \(f_{\omega}(f_2(2)) = f_{\omega}(f_{\omega}(2)) = f_{\omega}^2(2) = f_{\omega + 1}(2)\) The sequence can continue at this point with n = 3 to 7 and then p = 2 to 7. Work in progress 'Definitions' The definition of the function is accurately defined in program code (to be provided). A high level definition is as follows: \(J_2(n^n) = f_{(\omega\uparrow\uparrow(n-1))^{n+1}}(n+1)\) This is equivalent to the following simpler definition: \(J_2(n^n) = f_{\omega\uparrow\uparrow n}(n+1)\) but the longer definition serves a useful purpose by forcing the sequence of numbers such as: \(f_{\omega}(3) = f_3(3) = f_2^3(3)\) before \(f_2^3(4)\) and \(f_{\omega^{\omega}}(3) = f_{\omega^3}(3)\) before \(f_{\omega^3}(4)\) Work in progress 'Calculations' Here are some preliminary calculations for various values of r: \(J_2(0) = 0\) \(J_2(0.25) = 2\) \(J_2(0.5) = 4\) \(J_2(0.75) = 6\) \(J_2(1) = 8\) \(J_2(1.005) = f_2(3)\) \(J_2(1.045) = f_2(7)\) \(J_2(1.055) = f_2^2(3)\) \(J_2(1.38) = f_3^2(6)\) \(J_2(1.39) = f_3^2(7)\) \(J_2(1.405) = f_3^3(4)\) \(J_2(1.705) = f_4^3(5)\) \(J_2(2.305) = f_6^3(7)\) \(J_2(2.355) = f_6^4(7)\) \(J_2(2.405) = f_6^5(7)\) \(J_2(2.455) = f_6^6(7)\) \(J_2(2.5) = f_{\omega}(7)\) \(J_2(2.505) = f_{\omega}^2(6)\) \(J_2(2.51) = f_{\omega}^4(6)\) \(J_2(2.52) = f_{\omega+1}(6)\) \(J_2(2.53) = f_{\omega+1}^5(6)\) \(J_2(2.54) = f_{\omega+2}^2(5)\) \(J_2(2.55) = f_{\omega+2}^5(7)\) \(J_2(2.56) = f_{\omega+3}^3(4)\) \(J_2(2.57) = f_{\omega+3}^6(7)\) \(J_2(2.58) = f_{\omega+4}^3(7)\) \(J_2(2.6) = f_{\omega+5}^4(7)\) \(J_2(2.65) = f_{\omega.2+1}^3(4)\) \(J_2(2.7) = f_{\omega.2+4}^2(5)\) \(J_2(2.72) = f_{\omega.2+5}^2(7)\) \(J_2(2.725) = f_{\omega.2+5}^4(7)\) \(J_2(2.75) = f_{\omega.2+6}^6(7)\) \(J_2(2.8) = f_{\omega.3+2}^5(7)\) \(J_2(2.9) = f_{\omega.4+1}^3(6)\) \(J_2(3) = f_{\omega.4+6}^6(7)\) \(J_2(3.1) = f_{\omega.5+5}^4(7)\) \(J_2(3.2) = f_{\omega.6+4}^2(7)\) \(J_2(3.21) = f_{\omega.6+4}^5(7)\) \(J_2(3.22) = f_{\omega.6+5}^2(7)\) \(J_2(3.23) = f_{\omega.6+5}^6(7)\) \(J_2(3.24) = f_{\omega.6+6}^3(7)\) \(J_2(3.245) = f_{\omega.6+6}^5(7)\) \(J_2(3.25) = f_{\omega^2}(2)\) \(J_2(3.252) = f_{\omega^2}(3)\) \(J_2(3.255) = f_{\omega^2}(4)\) \(J_2(3.26) = f_{\omega^2}^2(2)\) \(J_2(3.27) = f_{\omega^2}^3(4)\) \(J_2(3.28) = f_{\omega^2}^4(5)\) \(J_2(3.29) = f_{\omega^2}^5(6)\) \(J_2(3.3) = f_{\omega^2}^6(7)\) \(J_2(3.32) = f_{\omega^2+1}^2(6)\) \(J_2(3.34) = f_{\omega^2+1}^5(6)\) \(J_2(3.36) = f_{\omega^2+2}(3)\) \(J_2(3.38) = f_{\omega^2+2}(6)\) \(J_2(3.4) = f_{\omega^2+2}^2(5)\) \(J_2(3.45) = f_{\omega^2+3}^5(6)\) \(J_2(3.5) = f_{\omega^2+4}^4(6)\) \(J_2(3.55) = f_{\omega^2+5}^4(6)\) \(J_2(3.6) = f_{\omega^2+6}^4(7)\) \(J_2(3.65) = f_{\omega^2.2}^2(3)\) \(J_2(3.7) = f_{\omega^2.2}^4(6)\) \(J_2(3.75) = f_{\omega^2.2}^6(7)\) \(J_2(3.8) = f_{\omega^2.2+1}^3(4)\) \(J_2(3.85) = f_{\omega^2.2+1}^5(6)\) \(J_2(3.9) = f_{\omega^2.2+2}(4)\) \(J_2(3.95) = f_{\omega^2.2+2}^2(3)\) \(J_2(4) = f_{\omega^2.2+2}^3(3)\) Work in progress Category:Blog posts